Optimal. Leaf size=33 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]
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Rubi [A] time = 0.0090743, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9}\\ \end{align*}
Mathematica [A] time = 0.0626643, size = 41, normalized size = 1.24 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 36, normalized size = 1.1 \begin{align*} -{\frac{-ex+d}{9\, \left ( ex+d \right ) ^{8}de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37059, size = 333, normalized size = 10.09 \begin{align*} -\frac{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5} +{\left (e^{4} x^{4} - 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{9 \,{\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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